Hi James,
Few things to clear up on optimization using WAAS.
1. The TCP session / connection has to pass thru both WAEs - server side and client side.
2. Central Manager - CM can not optimize. CM is just a Management portion of WAAS thru which you can configure WAAS using centralized Management GUI. More details here:
http://www.cisco.com/en/US/docs/app_ntwk_services/waas/waas/v441/configuration/guide/intro.html#wp1122501
Now to answer your questions:
Q1 : Tying together the networks above is a Cisco 3750 switch, it CAN function as a Multi-Layer switch. However, I do not see any WCCP redirection capable software for it.:
Ans: Please make sure you are using desktop routing template. CLI to verify this is: show sdm prefer.
Further WCCP configuration details can be found here:
https://www.cisco.com/en/US/docs/switches/lan/catalyst3750/software/release/12.2_52_se/configuration/guide/swwccp.html
Q2: Somehow this network worked once, then broke, unfortunately, I do not know the details of how it was functioning before it broke.
Ans: IT may be because of 3750 set t ovlan template which is default template or may be missing WCCP config on 3750. You also need WCCP config on WAAS side. Above link should help you with 3750 WCCP configuration adn for WAAS side WCCP configuration, please go thru this document:
http://www.cisco.com/en/US/docs/app_ntwk_services/waas/waas/v441/configuration/guide/traffic.html
Q3: Is there any way I can make this work WITHOUT the Core WAE? ie, WAE to be prepositioned talking directly to the file server through the CM?
Ans: Nope. You need WQAE devices on both sides and the TCP session must go thru both WAAS in order to optimize it
Hope this helps.
Regards.
PS: Please mark this as Answered, if this answers your question.